A Deep Learning approach to Reduced Order Modelling of Parameter Dependent Partial Differential Equations

TL;DR

This paper introduces a deep learning-based reduced order modeling technique for parameter-dependent PDEs, leveraging autoencoders to efficiently approximate solution manifolds, especially where traditional methods struggle due to slow decay in Kolmogorov n-width.

Contribution

It proposes a novel autoencoder-based approach for reduced order modeling, with theoretical analysis linking minimal latent dimension to the solution manifold's topology.

Findings

Outperforms classical POD-Galerkin models in complex PDE scenarios

Provides theoretical bounds on approximation errors and minimal latent dimensions

Demonstrates effectiveness on parametrized advection-diffusion PDEs with challenging features

Abstract

Within the framework of parameter dependent PDEs, we develop a constructive approach based on Deep Neural Networks for the efficient approximation of the parameter-to-solution map. The research is motivated by the limitations and drawbacks of state-of-the-art algorithms, such as the Reduced Basis method, when addressing problems that show a slow decay in the Kolmogorov n-width. Our work is based on the use of deep autoencoders, which we employ for encoding and decoding a high fidelity approximation of the solution manifold. To provide guidelines for the design of deep autoencoders, we consider a nonlinear version of the Kolmogorov n-width over which we base the concept of a minimal latent dimension. We show that the latter is intimately related to the topological properties of the solution manifold, and we provide theoretical results with particular emphasis on second order elliptic PDEs, characterizing the minimal dimension and the approximation errors of the proposed approach. The theory presented is further supported by numerical experiments, where we compare the proposed approach with classical POD-Galerkin reduced order models. In particular, we consider parametrized advection-diffusion PDEs, and we test the methodology in the presence of strong transport fields, singular terms and stochastic coefficients.